Properties

Label 4016.2.a.c
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 1
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + \beta_{2} q^{3} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} \) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{7} \) \( + ( -1 + \beta_{2} - \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{3} \) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{5} \) \( + ( 1 + \beta_{1} - \beta_{2} ) q^{7} \) \( + ( -1 + \beta_{2} - \beta_{3} ) q^{9} \) \( + ( 1 + \beta_{1} - \beta_{3} ) q^{11} \) \( + ( -3 - \beta_{2} + \beta_{3} ) q^{13} \) \(+ q^{15}\) \( + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} \) \( + ( 2 - 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} \) \( + ( -3 + \beta_{1} + 2 \beta_{3} ) q^{21} \) \( + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{23} \) \( + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{25} \) \( + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{27} \) \( + ( -1 - 2 \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{29} \) \( + ( 2 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{31} \) \( + ( -1 + \beta_{2} + \beta_{3} ) q^{33} \) \( + ( -2 + \beta_{1} + \beta_{2} ) q^{35} \) \( + ( -5 + 3 \beta_{1} + 2 \beta_{2} ) q^{37} \) \( + ( -2 + \beta_{1} - 4 \beta_{2} + \beta_{3} ) q^{39} \) \( + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{41} \) \( + ( 3 - 7 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} \) \( + ( 3 - 3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{45} \) \( + ( -4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{47} \) \( + ( -1 + \beta_{1} - 3 \beta_{3} ) q^{49} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{51} \) \( + ( 3 + \beta_{1} - 4 \beta_{3} ) q^{53} \) \( + ( - \beta_{1} + 2 \beta_{3} ) q^{55} \) \( + ( 5 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{57} \) \( + ( -5 + 2 \beta_{2} + 5 \beta_{3} ) q^{59} \) \( + ( -6 + \beta_{1} + \beta_{3} ) q^{61} \) \( + ( -4 + \beta_{3} ) q^{63} \) \( + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{65} \) \( + ( -8 + 5 \beta_{1} + 5 \beta_{2} ) q^{67} \) \( + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{69} \) \( + ( 2 - 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{71} \) \( + ( 3 - 4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{73} \) \( + ( -1 + \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{75} \) \( + ( 3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{77} \) \( + ( 6 + 7 \beta_{1} + 3 \beta_{2} - 8 \beta_{3} ) q^{79} \) \( + ( -2 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{81} \) \( + ( -1 + 9 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{83} \) \( + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{85} \) \( + ( -6 - \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{87} \) \( + ( -2 + \beta_{1} + 6 \beta_{3} ) q^{89} \) \( + ( -4 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{91} \) \( + ( -8 + \beta_{1} - \beta_{2} + 5 \beta_{3} ) q^{93} \) \( + ( -2 + 4 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} ) q^{95} \) \( + ( 5 + 2 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} ) q^{97} \) \( + ( -1 - 2 \beta_{1} + 2 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(4q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut -\mathstrut 12q^{13} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut q^{17} \) \(\mathstrut +\mathstrut 9q^{19} \) \(\mathstrut -\mathstrut 7q^{21} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 7q^{25} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut -\mathstrut 12q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 5q^{35} \) \(\mathstrut -\mathstrut 13q^{37} \) \(\mathstrut -\mathstrut 13q^{39} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut +\mathstrut 5q^{43} \) \(\mathstrut +\mathstrut 11q^{45} \) \(\mathstrut -\mathstrut 12q^{47} \) \(\mathstrut -\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 2q^{51} \) \(\mathstrut +\mathstrut 5q^{53} \) \(\mathstrut +\mathstrut 3q^{55} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 6q^{59} \) \(\mathstrut -\mathstrut 21q^{61} \) \(\mathstrut -\mathstrut 14q^{63} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut -\mathstrut 17q^{67} \) \(\mathstrut -\mathstrut 13q^{69} \) \(\mathstrut +\mathstrut 10q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 13q^{75} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 21q^{79} \) \(\mathstrut -\mathstrut 8q^{81} \) \(\mathstrut +\mathstrut q^{83} \) \(\mathstrut -\mathstrut 17q^{85} \) \(\mathstrut -\mathstrut 31q^{87} \) \(\mathstrut +\mathstrut 5q^{89} \) \(\mathstrut +\mathstrut 2q^{91} \) \(\mathstrut -\mathstrut 23q^{93} \) \(\mathstrut -\mathstrut 12q^{95} \) \(\mathstrut +\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(3\) \(x^{2}\mathstrut +\mathstrut \) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.737640
−0.477260
2.09529
−1.35567
0 −1.19353 0 −0.837853 0 2.93117 0 −1.57549 0
1.2 0 −0.294963 0 −3.39026 0 0.817703 0 −2.91300 0
1.3 0 1.29496 0 0.772223 0 1.80033 0 −1.32307 0
1.4 0 2.19353 0 0.455887 0 −2.54920 0 1.81156 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{4} \) \(\mathstrut -\mathstrut 2 T_{3}^{3} \) \(\mathstrut -\mathstrut 2 T_{3}^{2} \) \(\mathstrut +\mathstrut 3 T_{3} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).